Particle in a Box
A quantum particle confined to a one-dimensional infinite potential well can only exist in discrete energy states. Visualize the wavefunctions, probability densities, and energy levels — then create superpositions and watch the probability density oscillate in time.
Energy Levels
Superposition State
How it works
The particle in a box (or infinite square well) is one of the simplest quantum mechanical systems. A particle of mass m is confined to a region of length L by infinitely high potential walls. The particle cannot exist outside the box, so the wavefunction must be zero at the boundaries.
This boundary condition forces the wavefunction into standing-wave patterns: ψn(x) = √(2/L) sin(nπx/L). Only specific wavelengths fit, which means only discrete energy levels are allowed: En = n²π²ℏ² / (2mL²). Energy is proportional to n², so the spacing between levels grows with n.
Each eigenstate has n−1 nodes (zeros inside the box). The probability density |ψn(x)|² tells us where the particle is likely to be found, following Born’s rule. For a single eigenstate, the probability density is static — it doesn’t change with time.
However, when you create a superposition of different energy levels, the probability density oscillates. Each component accumulates phase at a rate proportional to its energy: ψ(x,t) = Σ cn ψn(x) e−iEnt/ℏ. The interference between components at different frequencies produces a probability density that sloshes back and forth — the quantum analog of a classical particle bouncing between walls.